The present volume continues the theory of series begun in Volume II,
and then proceeds to the theory of measurement. Geometry we have
found it necessary to reserve for a separate final volume,

In the theory of well-ordered series and compact series, we have followed
Cantor closely, except in dealing with Zermelo’s theorem (#257 — 8), and in
cases where Cantor’s work tacitly assumes the multiplicative axiom. Thus
what novelty there is, is in the main negative. In particular, the multi-
plicative axiom is required in all known proofs of the fundamental proposition
that the limit of a progression of ordinals of the second class (i.e. applicable
to series whose fields have N terms) is an ordinal of the second class (cf. *265).
In consequence of this fact, a very large part of the recognized theory of
transfinite ordinals must be considered doubtful.

Part VI, on the theory of ratio and measurement, on the other hand,
is new, though it is a development of the method initiated in Euclid Book V
and continued by Burali-Forti* Among other points in our treatment of
quantity to which we wish to draw attention we may mention the following.
(1) We regard our quantities as in a generalized sense “vectors,” and
therefore we regard ratios as holding’ between relations. (2) The hypothesis
that the vectors concerned in any context form a group, which has generally
been made prominent in such investigations, sinks with us into a very
subordinate position, being sometimes not verified at all, and at oth^r times
a consequence of other more fruitful hypotheses. (3) We have developed
a theory of ratios and real numbers which is prior to our theory of measure-
ment, and yet is not purely arithmetical, i.e. does not treat ratios as mere
couples of integers, but as relations between actual quantities such as two
distances or two periods of time. (4) In our theory of “vector families,”
which are families of the kind to which some form of measurement is